Problem: What is the extraneous solution to these equations? $\dfrac{x^2 + 22}{x + 4} = \dfrac{2x + 46}{x + 4}$
Answer: Multiply both sides by $x + 4$ $ \dfrac{x^2 + 22}{x + 4} (x + 4) = \dfrac{2x + 46}{x + 4} (x + 4)$ $ x^2 + 22 = 2x + 46$ Subtract $2x + 46$ from both sides: $ x^2 + 22 - (2x + 46) = 2x + 46 - (2x + 46)$ $ x^2 + 22 - 2x - 46 = 0$ $ x^2 - 24 - 2x = 0$ Factor the expression: $ (x + 4)(x - 6) = 0$ Therefore $x = -4$ or $x = 6$ At $x = -4$ , the denominator of the original expression is 0. Since the expression is undefined at $x = -4$, it is an extraneous solution.